Limit formulas for norms of tensor power operators

Given an operator $\phi :X\to Y$ between Banach spaces, we consider its tensor powers ${\phi }^{\otimes k}$ as operators from the $k$-fold injective tensor product of $X$ to the $k$-fold projective tensor product of $Y$. We show that after taking the $k$th root, the operator norm of ${\phi }^{\otimes k}$ converges to the $2$-dominated norm ${\gamma }_2^{\ast }(\phi )$, one of the standard operator ideal norms.